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E39 (1997 - 2003) The BMW 5-Series (E39 chassis) was introduced in the United States as a 1997 model year car and lasted until the 2004 when the E60 chassis was released. The United States saw several variations including the 525i, 528i, 530i and 540i. -- View the E39 Wiki

If the car was driven a tad spirited in a hot day, or really spirited in a cooler day, the aux fan might come on. The engine is really hot, and there is no coolant flow. The car still "awake", so it reads the temps. Once the temp of dual temp sensor goes above a certain range (I believe it's 98°C on the i6), it will trigger the aux fan. This can be tested, after turning the engine off, leave the key in posII and unlock OBD - KTMP. You will be surprised.

P.S.: Actually if the temp differential between the dual temp sensor and the radiator outlet temp sensor is smaller than x.xx°C (I am not sure what the value is), then the aux fan is triggered.
If the temp sensor that reads the engine temp is bust (in the dual temp sensor), or if the rad outlet temp sensor is bust, the aux fan will never turn on, and also the electronically controlled t-stat will be never opened more than the max opening in standard mode. The DME thinks the coolant is not hot enough.
Found out about this the hard way.

When I had my cooling system overhaul performed, by a BMW master tech, I provided two boxes of parts..he was impressed with the thoroughness of the parts I supplied (I had every hose, pulley, tensioner, thermostat, water pump, fan clutch, etc, etc)...but he added in the temp sensors, saying it was a good idea to replace them as well.

If the pusher fan stays on after a spirited driving, it's normal. The engine is a tad hotter than when it's driven under "normal" conditions.
If the pusher fan comes on after the car was driven "normal", then chances are the fan clutch is failing.
If the pusher fan never comes on, even if the KTMP shows more than 98°C, but comes on when you turn A/C on, the dual temp sensor is toast, and the car might even overheat (this is if the pusher fan relay checks OK with the GT1).

I have a brand new fan clutch installed.
(The one for the main cooling fan in the shroud)

"Burning 2nd is a lot like vitamin tonic. Overly harsh, tastes like crap, but somewhere in all that there's good intent......just have to learn to read between the lines, actually you have to squint really hard to see the good, but its there somewhere"

I have a brand new fan clutch installed.
(The one for the main cooling fan in the shroud)

If it's a Behr fan clutch made in CPR, google the issue with this brand. You will see, even new most of the time it's not doing the job properly a.k.a. pulling minimal air. Lots of e46 & e39 guys on other forums recognized the issue (especially the e46 guys) and the consensus is to run the Sachs fan clutch, which is very strong. I went this route a few years ago, and I am happy.

I will add this information to this thread on what brands to buy:
- What BMW E39 parts & supplies are most often recommended to buy OEM from a dealer or sponsor (1)

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Please read the suggested threads, where the best always add value to those threads, either by pictures or by descriptions, so the next person with the same problem stands on your shoulders.
See also: E39 Bestlinks & How to easily find what you need

If it's a Behr fan clutch made in CPR, google the issue with this brand. You will see, even new most of the time it's not doing the job properly a.k.a. pulling minimal air. Lots of e46 & e39 guys on other forums recognized the issue (especially the e46 guys) and the consensus is to run the Sachs fan clutch, which is very strong. I went this route a few years ago, and I am happy.